WASHINGTON — When President Barack Obama dined with Mexico’s President Enrique Pena Nieto this month at the Mexican leader’s official residence, the meal started with “laminas de atun,” thin slices of tuna.

The appetizer was not a surprising choice. Mexico has tried to get its yellowfin tuna on American plates for decades. Its fishermen are essentially frozen out of the lucrative U.S. market because they catch tuna with a method that has led to the demise of millions of dolphins, and falls below a standard U.S. officials set as “dolphin safe.”

But in recent months, Mexico has made progress in convincing the world that it is being treated unfairly because the U.S. tuna fishing regulation is not applied uniformly.

Mexico’s challenge is an attempt to increase its $7.5 million share of a U.S. tuna import market worth more than a half-billion dollars. But it also raises questions for U.S. consumers about whether the tuna they eat is truly “dolphin safe” — not sold at the expense of a mammal Americans cherish.

There is no sure way to catch tuna without harming other marine life. Dolphins, as well as sharks, turtles and other animals, are unintentionally killed as bycatch in the quest for tuna.

The central question facing governments, corporations, environmentalists and consumers is how much is too much, and whether using a huge net to catch tuna in one part of the ocean is any worse than using them to catch it in other parts.

The World Trade Organization recently agreed with Mexico’s claim that U.S. regulations in the eastern tropical Pacific Ocean, where the Mexican fleet fishes, are far more restrictive than they are for the western and central Pacific where the U.S. fleet fishes.

In response to the WTO ruling, the United States proposed a new rule to strengthen protections for dolphins wherever tuna is fished. The comment phase for the rule closed last week.

The proposal, drafted by the National Oceanic and Atmospheric Administration, has one change that irks Mexico — allowing captains in the western and central Pacific to self-certify they are not taking dolphins as bycatch.

“The objective . . . was to assure no dolphins were injured, and you’re not doing that,” said Mark Robertson, president of Potomac Global Advisors, which advises companies and governments in international disputes. “How practical is it to trust captains to say how many dolphins they harmed?”

Robertson said the solution is simple: require independent, trained observers on the largest ships that fish tuna, as Mexico does, to confirm dolphins are not being harmed anywhere in the ocean and assure that the dolphin-safe label on tuna sold in the United States is valid.

No place in the ocean is like the eastern tropical Pacific, where for reasons that marine biologists don’t fully understand, tuna and dolphins swim together.

“Marine mammals interact with most fishing gear only incidentally, but in the ETP tuna fishery, the dolphins are an intrinsic part of the fishing operation,” according to NOAA’s Southwest Fisheries Science Center.

“The fishermen intentionally capture both tuna and dolphins together, then release the dolphins from the net,” the center explained in a statement. “The bycatch of dolphins in the eastern tropical Pacific (ETP) purse-seine tuna fishery stands apart from marine mammal bycatch in other fisheries, not only in scale but in the way the dolphins interact with the fishery.”

For several decades starting in the 1950s, an estimated 150,000 dolphins were killed each year by the world’s fleets, including those of the United States — “estimated to be over 6 million animals, the highest known for any fishery,” NOAA said.

Ship captains have used explosives and sonic noises to round up dolphins and tuna, then circle them with purse-seine nets, huge nets that fold around the prey and are drawn shut. The majority of the dolphins are released alive, NOAA said.

In 1988, however, a biologist filmed hundreds of dolphins dying in a purse-seine net from a Panamanian boat, touching off a worldwide boycott of tuna. Two years later, Starkist, Bumble Bee and Chicken of the Sea, the world’s largest tuna-canning companies, agreed to stop buying and selling tuna caught in purse-seine nets. Congress blocked tuna fished with purse-seine nets from the U.S. market.

In the western and central Pacific, a majority of ships also use purse-seine nets, but take fewer dolphins because they do not swim with tuna as often as in the eastern Pacific, NOAA said. The ships also have the more relaxed level of regulation that prompted Mexico to call foul, NOAA said.

About 15 percent of tuna fishing in that area is done by another method, longlines trailing ships with hundreds of baited hooks. The Pew Charitable Trust described longlines as “indiscriminate and wasteful gear” that also “catches and kills more than 80 types” of animals fishermen do not target. The bycatch includes endangered sea turtles, blue and white marlin and severely depleted western Atlantic bluefin tuna.

To gain a larger stake of the U.S. market, Mexico works harder than most countries, placing trained observers on larger ships to monitor the dolphin catch.

“We make fishermen do things that no other fishery imposes. Very few fisheries require 100 percent observer coverage,” Aguilar said. “It’s really frustrating because we have answered everything the agreement has called for.”

NOAA acknowledges that. The bycatch of dolphins in the ETP tuna fishery has been reduced by more than 99 percent, the agency said. However, NOAA added, “Even at the present level of about 1,000 dolphins per year, it remains among the largest documented cetacean bycatch in the world.”

Today the dolphin population in the eastern Pacific is struggling to recover.

“That’s the reason why the [eastern Pacific] is the focal point” of strong regulations, said Rodney McInnis, administrator of the southwest region of NOAA Fisheries. “In other oceans marine mammals are occasionally harmed, but it’s not the practice to intentionally set gear to capture dolphin. That’s why we treated it different from the rest of the world.”